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Ultimate Channel Capacity of Free-Space
Optical Communications
Jeffrey H. Shapiro, Saikat Guha, and Baris I. Erkmen
Research Laboratory of Electronics, Massachusetts Institute of Technology,
Cambridge, MA, 02139
[email protected], [email protected], [email protected]
The ultimate classical information capacity of multiple-spatial-mode,
wideband, optical communications in vacuum between soft-aperture
transmit and receive pupils is considered. The ultimate capacity is shown
to be achieved by coherent-state encoding and joint measurements over
entire codewords. This capacity is compared with the capacities realized
with the same encoding and homodyne or heterodyne detection, which
are single-channel-use measurements. Realistic background spectral
radiance values are used to obtain tight bounds on the capacity
of single-spatial-mode, narrowband 1.55-µm-wavelength free-space
c 2005 Optical
communications in the presence of background light. Society of America
OCIS codes: 060.4510, 060.1660
1.
INTRODUCTION
Ubiquitous, reliable, high data-rate communication—carried by electromagnetic
waves at microwave to optical frequencies—is an essential ingredient of our technological age. Information theory seeks to delineate the ultimate limits on reliable
communication that arise from the presence of noise and other disturbances, and to
establish means by which these limits can be approached in practical systems. The
mathematical foundation for this assessment of limits is Shannon’s noisy channel
coding theorem [1], which introduced the notion of channel capacity—the maximum
mutual information between a channel’s input and output—as the highest rate at
which error-free communication could be maintained. Textbook treatments of channel capacity [2, 3] study channel models—ranging from the binary symmetric channel’s digital abstraction to the additive white-Gaussian-noise channel’s idealization
of thermal-noise-limited waveform transmission—for which classical physics is the
underlying paradigm. Fundamentally, however, electromagnetic waves are quantum
mechanical, i.e., they are Boson fields [4, 5]. Moreover, high-sensitivity photodetection systems have long been limited by noises of quantum mechanical origin [6].
Thus it would seem that determining the ultimate limits on optical communication would necessarily involve an explicitly quantum analysis, but such has not
been the case. Nearly all work on the communication theory of optical channels—
viz., that done for systems with laser transmitters and either coherent-detection or
direct-detection receivers—uses semiclassical (shot-noise) models (see, e.g., [7, 8]).
Here, electromagnetic waves are taken to be classical entities, and the fundamental
noise is due to the random release of discrete charge carriers in the process of photodetection. Inasmuch as the quantitative results obtained from shot-noise analyses
of such systems are known to coincide with those derived in rigorous quantummechanical treatments [9], it might be hoped that the semiclassical approach would
suffice. But, Helstrom’s derivation [10] of the optimum quantum receiver for binary
1
coherent-state (laser light) signaling demonstrated that the lowest error probability,
at constant average photon number, required a receiver that was neither coherent
detection nor direct detection. That Dolinar [11] was able to show how Helstrom’s
optimum receiver could be realized with a photodetection feedback system which
admits to a semiclassical analysis did not alleviate the need for a fully quantummechanical theory of optical communication, as Shapiro et al. [12] soon proved that
even better binary-communication performance could be obtained by use of twophoton coherent state (now known as squeezed state) light, for which semiclassical
photodetection theory did not apply.
In quantum mechanics, the state of a physical system together with the measurement that is made on that system determine the statistics of the outcome of that
measurement, see, e.g., [13]. Thus in seeking the classical information capacity of a
Bosonic channel, we must allow for optimization over both the transmitted quantum
states and the receiver’s quantum measurement. In particular, it is not appropriate
to immediately restrict consideration to coherent-state transmitters and coherentdetection or direct-detection receivers. Imposing these structural constraints leads
to Gaussian-noise (Shannon-type) capacity formulas for coherent (homodyne and
heterodyne) detection [14] and a variety of Poisson-noise capacity results (depending
on the power and bandwidth constraints that are enforced) for shot-noise-limited
direct detection [15]–[18]. None of these results, however, can be regarded as specifying the ultimate limit on reliable communication at optical frequencies. What is
needed for deducing the fundamental limits on optical communication is the analog of Shannon’s noisy channel coding theorem—unfettered by unjustified structural constraints on the transmitter and receiver—that applies to transmission of
classical information over a noisy quantum channel, viz., the Holevo-SchumacherWestmoreland (HSW) theorem [19]–[21].
Recently [22], we have used the HSW theorem to find the classical capacity of
the pure-loss Bosonic channel—in which signal photons may be lost in propagation, with the channel injecting the minimum quantum noise required to preserve
the Heisenberg uncertainty principle—and showed that it is achieved by singleuse encoding over coherent states with joint measurements over entire codewords.
Single-use coherent-state encoding was also used to obtain a lower bound on the
capacity of the thermal-noise channel—a lossy channel in which noise photons are
injected from the external environment—which we have shown to be tight in the
limit of low and high noise levels [23, 24]. Furthermore, if our conjecture concerning
the minimum output entropy of this isotropic-Gaussian noise channel is correct [25],
then single-use coherent-state encoding is capacity achieving.
Line-of-sight optical propagation between a finite-area transmitter exit pupil and
a finite-area receiver entrance pupil constitutes a lossy Bosonic channel. The vacuum
propagation case—whose fundamental propagation loss is due to diffraction—may
approach the pure-loss (minimum noise-injection) channel. With atmospheric propagation there will be additional losses arising from absorption and scattering [26],
as well as time-dependent fading resulting from atmospheric turbulence [27]. In
addition, appreciable extraneous (background) light collection may occur in this
case.
This paper addresses the ultimate limits on free-space optical communications by
determining the capacity of the multiple spatial mode, wideband, pure-loss channel. Realistic background spectral radiance values are then used to obtain tight
bounds on the capacity of single-spatial-mode, narrowband 1.55-µm-wavelength
free-space communications in the presence of background light. In both cases—
the wideband pure-loss channel, and the narrowband 1.55-µm-wavelength channel
2
with background light—we show that channel capacity is achieved with single-use
coherent-state encoding plus joint measurements over entire codewords, and we
quantify the capacity lost when coherent (homodyne or heterodyne) detection is
used in lieu of optimum reception. We begin our development with a brief review
of prior classical capacity results for lossy Bosonic channels.
2.
LOSSY BOSONIC CHANNELS
Although we are interested in classical communication over multi-mode lossy
Bosonic channels, it is convenient to begin with a treatment at the single-mode
level. In this case the channel input is an electromagnetic-field mode with annihilation operator â, and its output is another field mode with annihilation operator
â′ . The descriptions of this channel when multiple temporal and/or spatial modes
are employed can be built up from tensor-product constructions using the singlemode model. Neither the single-mode nor the multi-mode lossy channels constitute
unitary evolutions, so they are governed by trace-preserving completely-positive
(TPCP) maps [28] that relate their output density operators, ρ̂′ , to their input
density operators, ρ̂.
The TPCP map EηN (·) for the single-mode lossy channel can be derived from
the commutator-preserving beam splitter relation
p
√
(1)
â′ = η â + 1 − η b̂,
in which the annihilation operator b̂ is associated with an environmental (noise)
mode, and 0 < η < 1 is the channel transmissivity. For the pure-loss channel, the b̂
mode is in its vacuum state; for the thermal-noise channel this mode is in a thermal
state, viz., an isotropic-Gaussian mixture of coherent states with average photon
number N > 0,
Z
exp(−|β|2 /N )
ρ̂b = d2 β
|βihβ|.
(2)
πN
The classical capacity of the single-mode lossy channel is established by random
coding arguments akin to those employed in classical information theory. A set of
symbols {j} is represented by a collection of input states {ρ̂j } that are selected
according to some prior distribution {pj }. The output states {ρ̂′j } are obtained
by applying the channel’s TPCP map EηN (·) to these input symbols. The Holevo
information associated with priors {pj } and states {σˆj } is given by,


X
X
pj S(σ̂j ),
(3)
pj σ̂j  −
χ(pj , σ̂j ) = S
j
j
where S(σ̂) ≡ −tr(σ̂ ln(σ̂)) is the von Neumann entropy. According to the HolevoSchumacher-Westmoreland theorem [19]–[21], the capacity of this channel, in nats
per use, is
C = sup(Cn /n) = sup{ max [χ(pj , (EηN )⊗n (ρ̂j ))/n]},
n
n
{pj ,ρ̂j }
(4)
where Cn is the capacity achieved when coding is performed over n-channel-use
symbols and the supremum over n is necessitated by the fact that channel capacity
may be superadditive.
We have previously shown that the capacity of the single-mode, pure-loss channel
whose transmitter is constrained to use no more than N̄ photons on average is [22]
C = g(η N̄ ) nats/use,
3
(5)
where
g(x) ≡ (x + 1) ln(x + 1) − x ln(x)
(6)
is the Shannon entropy of the Bose-Einstein probability distribution. This capacity
is achieved by single-use random coding over coherent states using an isotropic
Gaussian distribution which saturates the transmitter’s bound on average photon
number. [Note that the optimality of single-use encoding means that the capacity
of the single-mode pure-loss channel is not superadditive.] This capacity exceeds
what is achievable with homodyne and heterodyne detection,
Chom =
1
ln(1 + 4η N̄ ) and Chet = ln(1 + η N̄ ),
2
(7)
respectively, although heterodyne detection is asymptotically optimal as N̄ → ∞.
An analytical expression for the direct-detection capacity corresponding to this
single-mode case is not known, but this capacity has been shown to satisfy [29],
Cdir ≤
1
ln(η N̄ ) + o(1)
2
and
lim (Cdir ) =
N̄ →∞
1
ln(η N̄ ),
2
(8)
and so is dominated by (5) for ln(η N̄ ) > 1.
For the pure-loss scalar channel in which the transmitter may use all frequencies
ω ∈ [0, ∞) of a single electromagnetic polarization subject to an average power
constraint P with all frequencies having the same channel transmissivity, we have
shown that the resulting channel capacity is [22]
r
πηP
nats/sec,
(9)
CWB =
3h̄
√
which is π/ 3 times higher than what can be achieved with homodyne or heterodyne detection. Once again, single-use encoding over a coherent-state ensemble
is employed, with low frequencies being used preferentially because of the average
power constraint. As yet, there is no corresponding wideband capacity result for
direct detection, because existing results [15, 17] ignore the frequency dependence
of photon energy by constraining photon flux rather than power.
For the thermal-noise channel, i.e., the lossy Bosonic channel with isotropicGaussian excess noise, we have obtained bounds on the channel capacity. For the
sake of brevity, we will restrict our discussion to the single-mode case. A lower
bound on the single-mode capacity for this channel is easily obtained. We assume
coherent-state encoding over single channel uses with an isotropic Gaussian prior
distribution. It then follows that
C ≥ g(η N̄ + (1 − η)N ) − g((1 − η)N ).
(10)
We believe that this single-use coherent-state encoding with an isotropic Gaussian
prior achieves channel capacity for the thermal-noise channel, i.e., we believe that
the right-hand side of Eq. (10) gives the capacity of this channel [23, 24]. Because
of the following upper bound on the single-mode channel capacity
X
pj S(ρ̂′j ) (11)
Cn /n ≤ max (S(ρ̂′ )/n) − min(S(ρ̂′j )/n), where ρ̂′ ≡
{pj ,ρ̂j }
=
ρ̂j
g(η N̄ + (1 − η)N ) −
j
min(S(ρ̂′j )/n),
ρ̂j
4
(12)
the proof of our conjecture is intimately related to the problem of determining the
minimum von Neumann entropy that can be realized at the output of the thermalnoise channel by choice of its input state [25]. So far, among many other things,
we have shown that a coherent-state input leads to a local minimum in the output
entropy, and we have shown that a coherent-state input minimizes the integer-order
Rényi output entropies [30, 31]. A proof of our capacity conjecture would follow immediately from the latter result were a rigorous foundation available for the replica
method of statistical mechanics. [The replica method has recently been applied to
other problems in communication theory [32, 33], so establishing its rigorous basis would have additional import outside of statistical physics and Bosonic-channel
communications.] Further support for our output entropy and capacity conjectures
comes from the suite of lower bounds that we have obtained on the thermal-noise
channel’s single-use output entropy [25]. These bounds provide fairly tight constraints on any possible gap between the channel’s minimum output entropy and
the associated coherent-state upper bound on this quantity. Indeed these results imply that coherent-state encoding approaches the C1 capacity at both low and high
noise levels. We have also developed numerical results that favor an even stronger
conjecture, viz., that the output states resulting from coherent-state inputs to the
thermal-noise channel majorize the output states arising from all other inputs [25].
This majorization conjecture, if true, would immediately imply both the minimum
output entropy and the capacity conjectures for the thermal-noise channel.
3.
MULTIPLE-SPATIAL-MODE, PURE-LOSS, FREE-SPACE CHANNEL
Although it serves a useful illustrative purpose, the wideband pure-loss channel
with frequency-independent loss is not a realistic scenario. Thus we have also studied the far-field, scalar free-space channel in which line-of-sight propagation of a
single polarization occurs over an L-m-long path from a circular transmitter pupil
(area At ) to a circular receiver pupil (area Ar ) with the
√ transmitter restricted
to use frequencies { ω : 0 ≤ ω ≤ ωc ≪ ω0 ≡ 2πcL/ At Ar }. This frequency
range is the far-field power transfer regime, wherein there is only a single spatial
mode that couples appreciable power from the transmitter pupil to the receiver
pupil, and its transmissivity at frequency ω is η(ω) = (ω/ω0 )2 ≪ 1. Figure 1
shows the geometry, the power allocations versus frequency for heterodyne, homodyne, and optimal reception, and their corresponding capacities versus normalized
power, P0 ≡ 2πh̄c2 L2 /At Ar , when only this dominant spatial mode is employed
[22]. Because far-field, free-space transmissivity increases as ω 2 , high frequencies
are used preferentially for this channel—unlike the case for frequency-independent
loss—because the transmissivity advantage of high-frequency photons more than
compensates for their higher energy consumption.
We have also explored the near-field behavior of the pure-loss free-space channel [23], by employing the full prolate-spheroidal wave function normal-mode decomposition associated with the propagation geometry shown in Fig. 1(a) [34, 35].
Near-field propagation at frequency ω = 2πc/λ prevails when Df = At Ar /(λL)2 ,
the product of the transmitter and receiver Fresnel numbers, is much greater than
unity. In this case there are approximately Df spatial modes with near-unity transmissivities, with all other modes affording insignificant power transfer from the
transmitter pupil to the receiver pupil. In what follows we shall take another approach to the wideband capacity of the pure-loss free-space channel, by employing either the Hermite-Gaussian (HG) or Laguerre-Gaussian (LG) mode sets that
are associated with the soft-aperture (Gaussian-attenuation pupil) version of the
Fig. 1(a) propagation geometry. Two benefits will be derived from this approach.
5
L
P/P0 = 3
(bits)
Ar
2πCWB /ω0
(c)
(b)
At
ω N̄ (ω)η(ωc )/ωc
(a)
ω/ωc
CWB −→
↑
CWB−hom
←− CWB−het
P/P0
Fig. 1. Capacity results for the far-field, free-space, pure-loss channel: (a) propagation geometry; (b) capacity-achieving power allocations h̄ω N̄ (ω) versus frequency ω for heterodyne (dashed curve), homodyne (dotted curve), and optimal
reception (solid curve), with ωc and h̄ωc /η(ωc ) being used to normalize the frequency and the power-spectra axes, respectively; and (c) wideband capacities of
optimal, homodyne, and heterodyne reception versus transmitter power P , with
P0 ≡ 2πh̄c2 L2 /At Ar used for the reference power.
First, closed-form expressions become available for the modal transmissivities, as
opposed to the hard-aperture case [Fig. 1(a)], for which numerical evaluations or
analytical approximations must be employed. Second, the LG modes have been the
subject of a great deal of interest, in the quantum optics and quantum information
communities [36], owing to their carrying orbital angular momentum. Thus it is
germane to explore whether they confer any special advantage in regards to classical information transmission. As we shall see, in the next subsection, the modal
transmissivities of the LG modes are isomorphic to those of the HG modes. Inasmuch as the latter do not convey orbital angular momentum, it is clear that such
conveyance is not essential to capacity-achieving classical communication over the
pure-loss free-space channel.
3.A. Propagation Model: Hermite-Gaussian and Laguerre-Gaussian Mode Sets
In lieu of the hard-aperture propagation geometry from Fig. 1(a), wherein the transmitter and receiver pupils are perfectly transmitting apertures within otherwise
opaque planar screens, we now introduce the soft-aperture propagation geometry
of Fig. 2. From the quantum version of scalar Fresnel diffraction theory [37], we
know that it is sufficient, insofar as this propagation geometry is concerned, to
identify a complete set of monochromatic spatial modes—for a single electromagnetic polarization of frequency ω = 2πc/λ = ck—that maintain their orthogonality
when transmitted through this channel. The resulting two mode sets—i.e., the mode
functions at the input and output of the Fig. 2 propagation geometry—constitute
a singular-value decomposition (SVD) of the linear propagation kernel (spatial impulse response) associated with this geometry, which we will now develop.
Let ui (~x ), for ~x a 2D vector in the transmitter’s exit-pupil plane, denote a
frequency-ω field entering the transmitter pupil that is normalized to satisfy
Z
d2 ~x |ui (~x )|2 = 1.
(13)
The resulting field that leaves the transmitter pupil is taken to be
uT (~x ) = exp(−|~x |2 /rT2 )ui (~x ),
6
(14)
which represents a soft-aperture (Gaussian-attenuation function) spatial truncation.
After free-space Fresnel diffraction over an L-m-long path, uT (~x ) produces a field
Z
exp(ikL + ik|~x − ~x ′ |2 /2L)
uR (~x ′ ) = d2 ~x uT (~x )
,
(15)
iλL
in the receiver’s entrance-pupil plane, where ~x ′ is a 2D vector in that plane. The
receiver employs a soft-aperture (Gaussian-attenuation function) entrance pupil, so
that the field immediately after this pupil is
2
uo (~x ′ ) = exp(−|~x ′ |2 /rR
)uR (~x ′ ).
Thus, the input-output (ui (~x )-to-uo (~x ′ )) relation for the Fig. 2 channel is
Z
′
uo (~x ) = d2 ~x ui (~x )h(~x ′ , ~x ),
(16)
(17)
where
2
)
h(~x ′ , ~x ) ≡ exp(−|~x ′ |2 /rR
exp(ikL + ik|~x − ~x ′ |2 /2L)
exp(−|~x |2 /rT2 ),
iλL
(18)
is the channel’s spatial impulse response.
Fig. 2. Propagation geometry with soft apertures.
The singular-value (normal-mode) decomposition of h(~x ′ , ~x ) is
h(~x ′ , ~x ) =
∞
X
√
ηm φm (~x ′ )Φ∗m (~x ),
(19)
m=1
where
1 ≥ η1 ≥ η2 ≥ η3 ≥ · · · ≥ 0,
(20)
are the modal transmissivities, {Φm (~x )} is a complete orthonormal (CON) set of
functions (input modes) on the transmitter’s exit-pupil plane, and {φm (~x ′ )} is a
CON set of functions (output modes) on the receiver’s entrance-pupil plane. Physically, this decomposition implies that h(~x ′ , ~x ) can be separated into a countablyinfinite set of parallel channels in which transmission of ui (~x ) = Φm (~x ) results
√
in reception of uo (~x ′ ) = ηm φm (~x ′ ). Singular-value decompositions are unique
7
if their {ηm } are distinct. When degeneracies exist—i.e., when there are multiple
modes with the same ηm value—the SVD is not unique. In particular, a linear
√
combination of input modes with the same ηm value produces ηm times that
same linear combination of the associated output modes after propagation through
h(~x ′ , ~x ). As we shall soon see, owing to singular-value degeneracies, the HG and
LG modes of the soft-aperture free-space channel are equivalent mode sets.
The spatial impulse response h(~x ′ , ~x ) has both rectangular and cylindrical symmetries. The Hermite-Gaussian modes provide an SVD of this channel that has
rectangular symmetry. With ~x = (x, y) in Cartesian coordinates, the HG input
modes are as follows:
!
!
√
√
√
2(1 + 4Df )1/4
2(1 + 4Df )1/4
2(1 + 4Df )1/4
Hn
x Hm
y
Φn,m (x, y) = √
rT
rT
rT π n! m! 2n+m
(1 + 4Df )1/2
k
2
2
× exp −
(x
+
y
)
,
+
i
rT2
2L
for n, m = 0, 1, 2, . . . , (21)
where Hp (·) is the pth Hermite polynomial and
Df =
2
krT2 krR
4L 4L
(22)
is the product of the transmitter-pupil and receiver-pupil Fresnel numbers for this
soft-aperture configuration. The modal transmissivities for the HG modes are
!n+m+1
p
1 + 2Df − 1 + 4Df
ηn,m =
,
(23)
2Df
and the HG output modes are
!
√
√
1/4
1/4
2(1
+
4D
2(1
+
4D
)
)
f
f
√
φn,m (x′ , y ′ ) =
Hn
x′
rR
in+m+1 rR π n! m! 2n+m
×
Hm
√
2(1 + 4Df )1/4 ′
y
rR
!
(1 + 4Df )1/2
k
′2
′2
exp −
(x + y ) ,
−i
2
rR
2L
for n, m = 0, 1, 2, . . . ,
(24)
where ~x ′ = (x′ , y ′ ). Because channel capacity depends only on the modal transmissivities, it is worth noting that
Z
Z
∞ X
∞
X
Df =
ηn,m = d2 ~x ′ d2 ~x |h(~x ′ , ~x )|2 ,
(25)
n=0 m=0
where the second equality is a consequence of (19) and the first equality can be
obtained either by summing the series or evaluating the double integral. Far-field
power transfer occurs when Df ≪ 1, in which case η0,0 ≈ Df and all the other modal
transmissivities are insignificantly small in comparison. Near-field power transfer occurs when Df ≫ 1, in which case there are many modes that couple appreciable
power from the transmitter pupil to the receiver pupil. However, because the HG
mode decomposition presumes soft-aperture pupils, the near-field modal transmissivities do not have the abrupt near-unity to near-zero transition that occurs for
the hard-aperture singular values.
8
The HG modes’ singular values have degeneracies, i.e., there are q HG modes
q
whose modal transmissivities equal η0,0
, hence the HG-mode SVD of h(~x ′ , ~x ) is not
unique. The Laguerre-Gaussian modes provide an alternative SVD for this channel,
one with cylindrical rather than rectangular symmetry. Using the polar coordinates
~x = (r, θ) we have that the LG input modes are
s
"√
#|ℓ|
(1 + 4Df )1/4
2 p!
2(1 + 4Df )1/4
Φp,ℓ (r, θ) =
r
π(|ℓ| + p)!
rT
rT
×
L|ℓ|
p
2(1 + 4Df )1/2 2
(1 + 4Df )1/2
k
2
r
+
iℓθ
,
exp
−
r
+
i
rT2
rT2
2L
for p = 0, 1, 2, . . ., and l = 0 ± 1, ±2, . . . ,
(26)
|ℓ|
where Lp (·) is the p|ℓ|th generalized Laguerre polynomial. The corresponding
modal transmissivities are given by
!(2p+|ℓ|+1)
p
1 + 2Df − 1 + 4Df
,
(27)
ηp,ℓ =
2Df
from which it can be seen that the HG modes with n + m + 1 = q span the same
eigenspace as the LG modes with 2p+ |ℓ|+ 1 = q, and hence are related by a unitary
transformation. The LG output modes are
s
#|ℓ|
"√
(1 + 4Df )1/4
2 p!
2(1 + 4Df )1/4 ′
′ ′
r
φp,ℓ (r , θ ) =
π(|ℓ| + p)! i2p+|ℓ|+1 rR
rR
×
L|ℓ|
p
2(1 + 4Df )1/2 ′2
(1 + 4Df )1/2
k
′2
′
r + iℓθ ,
exp −
r
−i
2
2
rR
rR
2L
for p = 0, 1, 2, . . ., and l = 0 ± 1, ±2, . . . ,
(28)
where ~x ′ = (r′ , θ′ ). As was the case for the HG modes, channel capacity when LG
modes are employed depends only on the modal transmissivities.
A single frequency-ω photon in the LG mode Φp,l (r, θ) carries orbital angular
momentum h̄ℓ directed along the propagation (z) axis, whereas that same photon
in the HG mode Φn,m (x, y) carries no z-directed orbital angular momentum. The
equivalence of the {ηp,l } and the {ηn,m } then implies that angular momentum does
not play a role in determining the ultimate—channel capacity—limit on classical
information transmission over the free-space channel shown in Fig. 2.
3.B. Wideband Capacities with Multiple Spatial Modes
Here we shall address the wideband capacities that can be achieved over the pureloss, scalar free-space channel shown in Fig. 2 using either heterodyne detection,
homodyne detection, or optimum (joint measurement over entire codewords) reception. We will allow the transmitter to use multiple spatial modes—from either the
HG or LG mode sets—and all frequencies ω ∈ [0, ∞) subject to a constraint, P , on
the average power in the field entering the transmitter’s exit pupil. It follows from
our prior work [22, 23] that the capacities we are seeking satisfy,
Z ∞
∞
X
dω
C(P ) = max
q
CSM (η(ω)q , N̄q (ω)),
(29)
2π
N̄q (ω)
0
q=1
9
where the maximization is subject to the average power constraint,
P =
Z
∞
X
q
0
q=1
and
q
η(ω) ≡
∞
dω
h̄ω N̄q (ω),
2π
!q
p
1 + 2(ω/ω0 )2 − 1 + 4(ω/ω0 )2
2(ω/ω0 )2
(30)
(31)
is the modal transmissivity at frequency ω with q-fold degeneracy, with ω0 =
4cL/rt rR being the frequency at which Df = 1. In (29),

g(η N̄ ),
for optimum reception



ln(1 + η N̄ ),
for heterodyne detection
CSM (η, N̄ ) ≡


 1
2 ln(1 + 4η N̄ ), for homodyne detection
(32)
are the relevant single-mode capacities as functions of the modal transmissivity, η,
and the average photon number, N̄ , for that mode. Regardless of the frequency
dependence of η(ω) the single-mode capacity formulas for heterodyne and homodyne detection imply that their wideband multiple-spatial-mode capacities bear the
following relationship,
1
(33)
Chom (P ) = Chet (4P ).
2
Thus, only two maximizations need to be performed—both of which can be done
via Lagrange multipliers—to obtain the wideband multiple-spatial-mode capacities
for optimum reception, heterodyne detection, and homodyne detection.
The results we have obtained by performing the preceding maximizations are as
follows. The optimum-reception capacity (in nats/sec) and its associated optimum
modal-power spectra are given by
∞
X
P
C(P ) =
−
q
h̄ω0 σ q=1
Z
and
∞
0
dω
ln[1 − exp(−ω/ω0 η(ω)q σ)],
2π
(34)
h̄ω/η(ω)q
,
exp(ω/ω0 η(ω)q σ) − 1
(35)
Z
∞
X
dω
ln(βω0 η(ω)q /ω),
q
2π
q=1
(36)
h̄ω N̄q (ω) =
respectively, where σ is a Lagrange multiplier chosen to enforce the average power
constraint. The corresponding capacity and optimum modal-power spectra for heterodyne detection are
Chet (P ) =
and
h̄ω N̄q (ω) = max[h̄ω0 (β − ω/ω0 η(ω)q ), 0],
(37)
where β is another Lagrange multiplier, again chosen to enforce the average power
constraint. We will omit the homodyne detection formulas, as they can be obtained
via scaling the heterodyne formulas according to (33), and instead focus our attention on the behavior of the capacity and power spectrum results.
10
A
B
Heterodyne: 6 spatial modes
Homodyne: 10 spatial modes
Optimum: all spatial modes
Fig. 3. (a) Capacity-achieving power spectra for wideband, multiple-spatial-mode
communication over the scalar, pure-loss, free-space channel when P = 8.12h̄ω02 .
Heterodyne detection (dashed curves) uses 6 spatial modes (from top to bottom, 1 ≤ q ≤ 3), homodyne detection (dot-dashed curves) uses 10 spatial modes
(from top to bottom, 1 ≤ q ≤ 4) and optimum reception (solid curves) uses all
spatial modes (although only top to bottom 1 ≤ q ≤ 6 are shown). (b) Wideband, multiple-spatial mode capacities for the scalar, pure-loss, free-space channel that are realized with optimum reception (solid curve), homodyne detection
(dot-dashed curve), and heterodyne detection (dashed curve). The capacities, in
bits/sec, are normalized by ω0 = 4cL/rT rR , the frequency at which Df = 1, and
plotted versus the average transmitter power normalized by h̄ω02 .
The capacity-achieving power spectrum for optimal reception employs all spatial
modes and all frequencies, whereas the capacity-achieving power spectra for heterodyne and homodyne detection are “water-filling” allocations, viz., they fill spatialmode/frequency volumes above their appropriate noise-to-transmissivity-ratio contours until the average power constraint is met. That water-filling power allocation
should be capacity achieving for these coherent detection cases is hardly a surprise,
as water-filling power allocation has long been known to be optimal for additive
Gaussian noise channels [38]. A consequence of water-filling power allocation is that
heterodyne and homodyne detection only employ a finite number of spatial modes
to achieve their respective capacities, whereas optimal-reception capacity needs all
spatial modes. This behavior is illustrated in Fig. 3(a), where we have plotted the
capacity-achieving power spectra for homodyne detection, heterodyne detection,
and optimal reception when P = 8.12h̄ω02 . In this case, heterodyne detection uses
1 ≤ q ≤ 3 (a total of 6 spatial modes) with non-zero power, and homodyne detection uses 1 ≤ q ≤ 4 (a total of 10 spatial modes) with non-zero power. Optimum
reception uses all spatial modes, but we have only plotted the spectra for 1 ≤ q ≤ 6.
In Fig. 3(b) we have plotted the heterodyne detection, homodyne detection, and
optimum reception capacities in bits/sec, normalized by ω0 , versus the normalized
power, P/h̄ω02 . Unlike the case seen in Fig. 1(c) for the wideband capacities of
the single-spatial-mode, far-field pure-loss channel—in which heterodyne detection
outperforms homodyne detection at high power levels—Fig. 3(b) shows that homodyne detection is consistently better than heterodyne detection for the multiplespatial-mode scenario. This behavior has a simple physical explanation. Consider
first the single-spatial mode wideband capacities. At low power levels, when capacity is power limited, homodyne detection outperforms heterodyne detection because
11
at every frequency it suffers less noise. On the other hand, at high enough power
levels single-spatial mode communication becomes bandwidth limited. In this case
heterodyne detection’s factor-of-two bandwidth advantage over homodyne detection
carries the day. Things are different when multiple spatial modes are available. In
this case, increasing power never reaches bandwidth-limited operation; additional,
lower transmissivity, spatial modes get employed as the power is increased so that
the noise advantage of homodyne detection continues to give a higher channel capacity than does heterodyne detection.
Figure 3 shows that the wideband capacity realized with optimum reception,
on the multiple-spatial-mode pure-loss channel, increasingly outstrips that of homodyne detection with increasing transmitter power. This advantage indicates that
joint measurements over entire codewords—which are implicit in the Holevo information maximization procedure that leads to the optimum-reception capacity—
afford performance that is unapproachable with homodyne detection, which is a
single-use quantum measurement.
4.
SINGLE SPATIAL-MODE COMMUNICATION AT 1.55 µm
The wideband results from the previous section set ultimate limits on free-space
optical communications in the absence of atmospheric effects. In this final section,
we consider a more constrained situation that will better approximate what can
be accomplished over an atmospheric path. Specifically, we consider single-spatialmode propagation at the fiber-compatible 1.55-µm-wavelength in the presence of
atmospheric extinction and background light. The propagation geometry will be
the soft-aperture configuration shown in Fig. 2, but the spatial-impulse response at
the transmitter’s center frequency ω will now be taken to be
exp(−αL/2 + ikL + ik|~x − ~x ′ |2 /2L)
exp(−|~x |2 /rT2 ),
iλL
(38)
where α is the atmospheric extinction (absorption plus scattering) coefficient. The
transmitter will be assumed to be narrowband, i.e., its radian-frequency bandwidth,
Ω, is small enough that the energy of all transmitted photons is approximately
h̄ω. We shall assume that the transmitter uses the maximum-power-transfer HG
input mode (which is identical to the maximum-power-transfer LG input mode) and
that the receiver (whether it uses homodyne, heterodyne, or optimum reception)
measures the corresponding output mode. Because of the combination of diffraction
loss and extinction loss, the associated transmissivity for this atmospheric channel
is
!
p
1 + 2(ω/ω0 )2 − 1 + 4(ω/ω0 )2
ηa =
e−αL ,
(39)
2(ω/ω0 )2
2
h(~x ′ , ~x) = exp(−|~x ′ |2 /rR
)
which, by our narrowband assumption, is constant over the frequencies used by the
transmitter.
Accompanying the atmospheric propagation loss is the collection of extraneous
(background) light. For each temporal mode arriving at the receiver, this background light is an isotropic mixture of coherent states with average photon number
[39]
N = π106 λ3 Nλ /h̄ω 2 ,
(40)
where Nλ is the background spectral radiance (in W/m2 SR µm). A typical daytime
value Nλ ∼ 10 W/m2 SR µm at 1.55 µm then leads to N ∼ 10−6 ; nighttime Nλ
12
10
10
(b)
C(P ) (Tbits/s)
C(P ) (Tbits/s)
(a)
1
0.1
0.01
0.001
-60
1
0.1
0.01
-50
-40
-30
-20
-10
1
P (dBW)
L (km)
10
Fig. 4. Capacities of the single-spatial-mode, narrowband, 1.55-µm-wavelength
channel for optimum reception (solid curves), homodyne detection (dot-dashed
curves) and heterodyne detection (dashed curves) plotted in (a) versus average
transmitter power and in (b) versus path length. Both (a) and (b) assume Ω/2π =
1 THz, α = 0.5 dB/km, rT = 1 mm, and rR = 1 cm; (a) assumes L = 1 km, and
(b) assumes P = 1 mW.
values are several orders of magnitude lower [39]. Because the single-mode, thermalnoise channel EηN is the concatenation of the pure-loss channel Eη0 with a classicalnoise channel of average noise-photon number (1 − η)N [25], it follows that the
capacity of a thermal-noise channel can never exceed that of the pure-loss channel
with the same transmissivity. As result, the optimal-reception capacity of a single
temporal mode of our atmospheric channel satisfies
g(ηa N̄ + (1 − ηa )N ) − g((1 − ηa )N ) ≤ CSM (ηa , N̄ ) ≤ g(ηa N̄ ),
(41)
and these single-mode-capacity upper and lower bounds are virtually coincident for
ηa N̄ ≥ 10(1 − ηa )N . For the narrowband, multiple-temporal-mode case, we then
have
Ω
Ω
[g(ηa 2πP/h̄ωΩ + (1 − ηa )N ) − g((1 − ηa )N )] ≤ C(P ) ≤
g(ηa 2πP/h̄ωΩ). (42)
2π
2π
Thus, as long as ηa 2πP/h̄ωΩ ≥ 10(1 − ηa )N , these bounds are exceedingly tight,
and background light can be neglected in determining the capacity achieved with
optimum reception. The impact of the background light on the multiple-temporalmode channel capacities of homodyne and heterodyne detection is also negligible,
because these capacities are given by

Ω ln 1 + ηa 8πP/h̄ωΩ


for homodyne detection
 4π
1 + 2(1 − ηa )N
C(P ) =
(43)


 Ω ln 1 + ηa 2πP/h̄ωΩ
for
heterodyne
detection,
2π
1 + (1 − ηa )N
and N ≪ 1.
In Fig. 4 we have plotted the single-spatial-mode, narrowband, 1.55-µmwavelength capacities versus power [Fig. 4(a)] and path length [Fig. 4(b)]. Both
of these plots assume Ω/2π = 1 THz, α = 0.5 dB/km, rT = 1 mm, and rR = 1 cm.
Figure 4(a) assumes L = 1 km, and Fig. 4(b) assumes P = 1 mW. These numbers
represent clear-weather propagation—neglecting atmospheric turbulence—with a
13
∼1 mR transmitter-beam divergence, and transmitter powers commensurate with
semiconductor-laser/power-amplifier technology. The curves show that optimum reception increasingly outstrips homodyne and heterodyne detection at lower transmitter powers and longer path lengths. The crossing of the heterodyne and homodyne capacity-versus-power curves is the same power-limited versus bandwidthlimited behavior we discussed in conjunction with single-spatial-mode, wideband,
far-field communication, as is the asymptotic approach of the heterodyne capacity
to the optimum-reception capacity with increasing transmitter power, cf. Fig. 1(c).
5.
DISCUSSION
We have drawn upon the quantum description of Bosonic communication to establish ultimate limits on classical communications at optical frequencies. Although
most of our results have addressed vacuum propagation, we showed how extinction and background noise can be incorporated into a single-spatial-mode capacity
analysis. The principal propagation effect omitted from our development is the
time-dependent fading incurred as a result of atmospheric turbulence. Some results
are available on the classical capacity of the turbulent channel, see, e.g., [40], [41],
but these studies have employed the semiclassical approach. Thus, a full quantum
treatment is still needed.
6.
ACKNOWLEDGMENTS
This work was supported by the DoD Multidisciplinary University Research Initiative (MURI) program administered by the Army Research Office under Grant
DAAD19–00-1–0177.
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